Week 2

• Central tendency
• Variability of data

Note on sampling

Sampling

• sampling is the process by which \(n\) observations are taken from a population of size \(N\)
• this is one of the most important methods in the behavioural and social sciences
• if the sampling is wrong, the rest is BS
• GIGO principle (garbage in, garbage out)
• more in week 4
• for now: sample = subset of the population

Part 1: Central tendency

• Aim: we want to describe the data
• specifically: we want to express the center of the data distribution
• remember: think of data = distribution

Example data

• We take a sample of \(n=100\) of students at TiU
• And ask: how many hours per week do you spend on YouTube?
• Answers in full hours

Looking at the histogram

Describing central tendency

The MODE:

• simple definition: the score (or category) with the highest frequency
• works for all scales of data (think about nominal data)

Obtaining the mode

We look at the frequency table, and select the most frequently chosen option:

The mode is 10 hours.

The location of the mode

The mode and distribution shapes

(demo)

Describing central tendency

The MEAN:

• often called the average
• exact definition: the sum of all scores divided by the number of scores

Statistical notation:

\(\mu=\frac{\sum{X}}{N}\) (population mean)

\(M=\frac{\sum{X}}{n}\) (sample mean)

Calculating the mean

• Sample size: \(n=5\)
• YouTube hours watched data: \(5,7,9,14,6\)

\(\sum{X} = 5+7+9+14+6 = 41\)

\(M=\frac{\sum{X}}{n} = \frac{41}{5} = 8.20\)

Where is it in the distribution?

Mode and mean

Why not always the mean?

Suppose there are 10 friends (a, b, c, … j) in a bar. Each of them says how many hours they spend on YouTube last week.

Here’s their data:

…

Now another person enters. This friend, “k”, is a binge watcher. He says that last week he watched 50 hours of YouTube.

What do you think will happen to the mean?

New histogram

Beware of outliers

• Mean before: \(M=\frac{\sum{X}}{n} = \frac{72}{10} = 7.20\)
• Mean with binge-watcher: \(M=\frac{\sum{X}}{n} = \frac{122}{11} = 11.09\)

Extreme values can affect the mean!

The extreme values are often called outliers.

Another illustration

There are a hundred people in a bar. The average (mean) income is 30,000 EUR. Now Jeff Bezos walks in and suddenly everyone is billionaire.

These problems can be addressed:

• mean trimming (not in this course)
• another metric

Describing central tendency

The MEDIAN:

• often called the midpoint
• exact definition: the median splits the distribution in half

Example

The friend data:

Obtaining the median

1. sorting the data

Obtaining the median

2. find the value that lies in the middle

Here: We know we have 11 values, so the 6th value has 5 points to its left and right.

Mean and median

Special cases

Distributions without “clear” midpoint:

• data: \(4,15,13,14,38,3\)
• sorted data: \(3,4,13,14,15,38\)
• median?

In this case, we take the two middle values and obtain the average:

• median = \(\frac{13+14}{2}=13.5\)

Part 2: Variability

• Aim: we want to describe the data
• specifically: we want to express how much the scores in the data differ
• also called the spread of the data (or lack thereof)

New data example

• grades for Intro to Statistics at first attempt for \(N=10\)

How can we express data variability?

• The easiest way: we take the lowest value and the highest value
• \(\min grade = 3\)
• \(\max grade = 10\)

\(range = \max - \min\)

See also p. 102 in the book.

A bit more nuanced

• maybe we calculate how much each score differs from the (population) mean
• \(\mu = 6.8\)

What is problematic?

This procedure gives us the deviation score (from the mean) for each value

\(deviation = X - \mu\)

• Think about what the mean actually is
• It is - by definition - the balancing point
• Have a look…

Deviation and the mean

Deviations sum to 0

Common trick: Squaring the difference

The \(x^2\) trick

• removes negative values
• “punishes” larger values
• \(2^2 = 4\)
• \(4^2 = 16\)
• Note: differences are also squared
• When we double \(x\), we quadruple \(x^2\)

From deviation to variance

We can obtain a more meaningful measure now.

The mean of squared deviations is called the variance.

\(var = \frac{\sum{(X-\mu)^2}}{N}\)

Stepwise: deviation

\(\mu = 5.4\)

Stepwise: squared deviation

\(var = \frac{\sum{(X-\mu)^2}}{N} = \frac{9.2}{5} = 1.84\)

Stepwise: the standard deviation

• among the most frequently used statistics for variability
• standard in most research papers

\(SD = \sqrt{var}\)

\(\sigma = \sqrt{\frac{\sum{(X-\mu)^2}}{N}}\)

Here: \(\sigma = \sqrt{\frac{9.2}{5}} = \sqrt{1.84} = 1.36\)

Sum of squares

• an alternative approach is to first go through the sum of squared deviations (SS)
• this: \(\sum{(X-\mu)^2}\)

Then:

\(var = \frac{SS}{N}\)

\(\sigma = \sqrt{\frac{SS}{N}}\)

This is why \(var\) is also noted as \(\sigma^2\)

Remember populations and samples?

Until here: the variability statistics were for the population

The sample is biased (i.e. over- or underestimated):

• here this means it will underestimate the variability of the population
• we can correct for this
• this is where we need the sum of squares

Correcting for bias

We make the value slightly larger, by decreasing the denominator:

\(sample\ variance = \frac{SS}{n-1}\)

\(s = \sqrt{\frac{SS}{n-1}}\)

Compare:

• \(\frac{SS}{N} = \frac{9.2}{5} = 1.84\) vs \(\frac{SS}{n-1} = \frac{9.2}{4} = 2.30\)
• \(\sqrt{\frac{9.2}{5}} = 1.36\) vs \(\sqrt{\frac{9.2}{4}} = 1.52\)

Examples of reporting summary statistics

show that the judgments are closer to the true emotion score in the longer texts (M=1.19, SD=1.88) than in the shorter ones (M=2.00, SD=2.35), Cohen’s d = 0.38 [99% CI: 0.30; 0.45]

Examples of reporting summary statistics

The temporal evolution of a far‑right forum

Recap

• we can describe the center of the data
  • mode
  • mean
  • median
• we can also describe how much the data is spread out
  • range
  • deviation –> variance –> standard deviation
  • correcting for sample bias in sample statistics

Next week

• probability
• z-scores